An example of a weak decay of a nucleus is (Z, A) → (Z +1, A) + τ- + ντ-bar where (Z,A) represents a nucleus with Z protons and A−Z neutrons.
(d) Suppose the gauge boson in the above process is produced at rest. Assuming the neutrino is massless, what must the 3-momenta of the τ- and neutrino be? [5 Marks]
(e) Now suppose that the neutrino is not massless. Express the mass of the gauge boson in terms of the lepton and neutrino masses and their 3-momenta. [3 Marks]
(f) Consider the case where the nucleus (Z,A) is at rest and the nucleus (Z +1,A) recoils with energy E along the X direction. The τ is observed to be moving along the Y direction (i.e. at right-angles to the nucleus’s recoil). What will the component of the 3-momenta of the neutrino be along the axis of the nucleus’s recoil? You should express your result in terms of E,Z,A,Mp and Mn. [7 Marks]
E before= E after
3-momenta before= 3-momenta after
4-momentum before= 4 momentum after
The Attempt at a Solution
(d) Isolating the W_ bit of the process: W-= τ- + ντ-bar
so (MW-, 0) = (Eτ_, Pτ_) + (Eντ, Pντ)
And from 3-momentum conservation, the momenta of the neutrino and taon must be equal and opposite.
(My first issue is that i'm not sure that I can just isolate this part of the interaction, I just made an assumption from the Feynman diagram I drew). I also don't understand the significance of the neutrino being massless other than the |kτ| = Eν(bar)τ ?
(e) Applying 4-momentum conservation: MW2 = (Eτ + Eν(bar)τ)2 - (K τ + K ν(bar)τ)2
But I can't seem to work this through to gain an answer in terms of lepton neutrino masses and their 3-momentum.
(f) I'm not sure how to even approach this part, I cant figure out whether I need to write the 4 momenta's using A-Z etc. Any kind of hint or guidance would be great so I can have a good stab.
Thanks in advance for any help, it's greatly appreciated!